SLIDE 1
1
AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY
Andrew Ranicki and Daniele Sepe (Edinburgh) http://www.maths.ed.ac.uk/aar Maslov index seminar, 9 November 2009
SLIDE 2 2 The 1-dimensional Lagrangians
◮ Definition (i) Let R2 have the symplectic form
[ , ] : R2 × R2 → R ; ((x1, y1), (x2, y2)) → x1y2 − x2y1 .
◮ (ii) A subspace L ⊂ R2 is Lagrangian of (R2, [ , ]) if
L = L⊥ = {x ∈ R2 | [x, y] = 0 for all y ∈ L} .
◮ Proposition A subspace L ⊂ R2 is a Lagrangian of (R2, [ , ]) if and
- nly if L is 1-dimensional,
◮ Definition (i) The 1-dimensional Lagrangian Grassmannian Λ(1) is
the space of Lagrangians L ⊂ (R2, [ , ]), i.e. the Grassmannian of 1-dimensional subspaces L ⊂ R2.
◮ (ii) For θ ∈ R let
L(θ) = {(rcos θ, rsin θ) | r ∈ R} ∈ Λ(1) be the Lagrangian with gradient tan θ.
SLIDE 3
3 The topology of Λ(1)
◮ Proposition The square function
det2 : Λ(1) → S1 ; L(θ) → e2iθ and the square root function ω : S1 → Λ(1) = R P1 ; e2iθ → L(θ) are inverse diffeomorphisms, and π1(Λ(1)) = π1(S1) = Z .
◮ Proof Every Lagrangian L in (R2, [ , ]) is of the type L(θ), and
L(θ) = L(θ′) if and only if θ′ − θ = kπ for some k ∈ Z . Thus there is a unique θ ∈ [0, π) such that L = L(θ). The loop ω : S1 → Λ(1) represents the generator ω = 1 ∈ π1(Λ(1)) = Z .
SLIDE 4
4 The Maslov index of a 1-dimensional Lagrangian
◮ Definition The Maslov index of a Lagrangian L = L(θ) in (R2, [ , ]) is
τ(L) = 1 − 2θ π if 0 < θ < π if θ = 0
◮ The Maslov index for Lagrangians in n
(R2, [ , ]) is reduced to the special case n = 1 by the diagonalization of unitary matrices.
◮ The formula for τ(L) has featured in many guises besides the Maslov
index (e.g. as assorted η-, γ-, ρ-invariants and an L2-signature) in the papers of Arnold (1967), Neumann (1978), Atiyah (1987), Cappell, Lee and Miller (1994), Bunke (1995), Nemethi (1995), Cochran, Orr and Teichner (2003), . . .
◮ See http://www.maths.ed.ac.uk/˜aar/maslov.htm for detailed
references.
SLIDE 5
5 The Maslov index of a pair of 1-dimensional Lagrangians
◮ Note The function τ : Λ(1) → R is not continuous. ◮ Examples τ(L(0)) = τ(L(π
2 )) = 0, τ(L(π 4 )) = 1 2 ∈ R with L(0) = R ⊕ 0 , L(π 2 ) = 0 ⊕ R , L(π 4 ) = {(x, x) | x ∈ R} .
◮ Definition The Maslov index of a pair of Lagrangians
(L1, L2) = (L(θ1), L(θ2)) in (R2, [ , ]) is τ(L1, L2) = − τ(L2, L1) = 1 − 2(θ2 − θ1) π if 0 θ1 < θ2 < π if θ1 = θ2.
◮ Examples τ(L) = τ(R ⊕ 0, L), τ(L, L) = 0.
SLIDE 6
6 The Maslov index of a triple of 1-dimensional Lagrangians
◮ Definition The Maslov index of a triple of Lagrangians
(L1, L2, L3) = (L(θ1), L(θ2), L(θ3)) in (R2, [ , ]) is τ(L1, L2, L3) = τ(L1, L2) + τ(L2, L3) + τ(L3, L1) ∈ {−1, 0, 1} ⊂ R .
◮ Example If 0 θ1 < θ2 < θ3 < π then τ(L1, L2, L3) = 1 ∈ Z. ◮ Example The Wall computation of the signature of C P2 is given in
terms of the Maslov index as σ(C P2) = τ(L(0), L(π/4), L(π/2)) = τ(L(0), L(π/4)) + τ(L(π/4), L(π/2)) + τ(L(π/2), L(0)) = 1 2 + 1 2 + 0 = 1 ∈ Z ⊂ R .
SLIDE 7 7 The Maslov index and the degree I.
◮ A pair of 1-dimensional Lagrangians (L1, L2) = (L(θ1), L(θ2))
determines a path in Λ(1) from L1 to L2 ω12 : I → Λ(1) ; t → L((1 − t)θ1 + tθ2) .
◮ For any L = L(θ) ∈ Λ(1)\{L1, L2}
(ω12)−1(L) = {t ∈ [0, 1] | L((1 − t)θ1 + tθ2) = L} = {t ∈ [0, 1] | (1 − t)θ1 + tθ2 = θ} = θ − θ1 θ2 − θ1
θ2 − θ1 < 1 ∅
◮ The degree of a loop ω : S1 → Λ(1) = S1 is the number of elements in
ω−1(L) for a generic L ∈ Λ(1). In the geometric applications the Maslov index counts the number of intersections of a curve in a Lagrangian manifold with the codimension 1 cycle of singular points.
SLIDE 8
8 The Maslov index and the degree II.
◮ Proposition A triple of Lagrangians (L1, L2, L3) determines a loop in
Λ(1) ω123 = ω12ω23ω31 : S1 → Λ(1) with homotopy class the Maslov index of the triple ω123 = τ(L1, L2, L3) ∈ {−1, 0, 1} ⊂ π1(Λ(1)) = Z .
◮ Proof It is sufficient to consider the special case
(L1, L2, L3) = (L(θ1), L(θ2), L(θ3)) with 0 θ1 < θ2 < θ3 < π, so that det2ω123 = 1 : S1 → S1 , degree(det2ω123) = 1 = τ(L1, L2, L3) ∈ Z
SLIDE 9 9 The Euclidean structure on R2n
◮ The phase space is the 2n-dimensional Euclidean space R2n, with
preferred basis {p1, p2, . . . , pn, q1, q2, . . . , qn}.
◮ The 2n-dimensional phase space carries 4 additional structures. ◮ Definition The Euclidean structure on R2n is the positive definite
symmetric form over R ( , ) : R2n × R2n → R ; (v, v′) →
n
xjx′
j + n
yky′
k ,
(v =
n
xjpj +
n
ykqk , v′ =
n
x′
jpj + n
y′
kqk ∈ R2n) . ◮ The automorphism group of (R2n, ( , )) is the orthogonal group
O(2n) of invertible 2n × 2n matrices A = (ajk) (ajk ∈ R) such that A∗A = I2n with A∗ = (akj) the transpose.
SLIDE 10 10 The complex structure on R2n
◮ Definition The complex structure on R2n is the linear map
ı : R2n → R2n ;
n
xjpj +
n
ykqk →
n
xjpj −
n
ykqk such that ı2 = −1. Use ı to regard R2n as an n-dimensional complex vector space, with an isomorphism R2n → Cn ; v → (x1 + iy1, x2 + iy2, . . . , xn + iyn) .
◮ The automorphism group of (R2n, ı) = Cn is the complex general
linear group GL(n, C) of invertible n × n matrices (ajk) (ajk ∈ C).
SLIDE 11 11 The symplectic structure on R2n
◮ Definition The symplectic structure on R2n is the symplectic form
[ , ] : R2n × R2n → R ; (v, v′) → [v, v′] = (ıv, v′) = − [v′, v] =
n
(x′
jyj − xjy′ j ) . ◮ The automorphism group of (R2n, [ , ]) is the symplectic group Sp(n)
- f invertible 2n × 2n matrices A = (ajk) (ajk ∈ R) such that
A∗ In −In
In −In
SLIDE 12 12 The n-dimensional Lagrangians
◮ Definition Given a finite-dimensional real vector space V with a
nonsingular symplectic form [ , ] : V × V → R let Λ(V ) be the set of Lagrangian subspaces L ⊂ V , with L = L⊥ = {x ∈ V | [x, y] = 0 ∈ R for all y ∈ L} .
◮ Terminology Λ(R2n) = Λ(n). The real and imaginary Lagrangians
Rn = {
n
xjpj | xj ∈ Rn} , ıRn = {
n
ykqk | yk ∈ Rn} ∈ Λ(n) are complementary, with R2n = Rn ⊕ ıRn.
◮ Definition The graph of a symmetric form (Rn, φ) is the Lagrangian
Γ(Rn,φ) = {(x, φ(x)) | x =
n
xjpj, φ(x) =
n
n
φjkxjqk} ∈ Λ(n) complementary to ıRn.
◮ Proposition Every Lagrangian complementary to ıRn is a graph.
SLIDE 13 13 The hermitian structure on R2n
◮ Definition The hermitian inner product on R2n is defined by
, : R2n × R2n → C ; (v, v′) → v, v′ = (v, v′) + i[v, v′] =
n
(xj + iyj)(x′
j − iy′ j ) .
, : Cn × Cn → C ; (z, z′) → z, z′ =
n
zjz′
j . ◮ The automorphism group of (Cn, , ) is the unitary group U(n) of
invertible n × n matrices A = (ajk) (ajk ∈ C) such that AA∗ = In, with A∗ = (akj) the conjugate transpose.
SLIDE 14
14 The general linear, orthogonal and unitary groups
◮ Proposition (Arnold, 1967) (i) The automorphism groups of R2n with
respect to the various structures are related by O(2n) ∩ GL(n, C) = GL(n, C) ∩ Sp(n) = Sp(n) ∩ O(2n) = U(n) .
◮ (ii) The determinant map det : U(n) → S1 is the projection of a fibre
bundle SU(n) → U(n) → S1 .
◮ (iii) Every A ∈ U(n) sends the standard Lagrangian ıRn of (R2n, [ , ])
to a Lagrangian A(ıRn). The unitary matrix A = (ajk) is such that A(ıRn) = ıRn if and only if each ajk ∈ R ⊂ C, with O(n) = {A ∈ U(n) | A(ıRn) = ıRn} ⊂ U(n) .
SLIDE 15
15 The Lagrangian Grassmannian Λ(n) I.
◮ Λ(n) is the space of all Lagrangians L ⊂ (R2n, [ , ]). ◮ Proposition (Arnold, 1967) The function
U(n)/O(n) → Λ(n) ; A → A(ıRn) is a diffeomorphism.
◮ Λ(n) is a compact manifold of dimension
dim Λ(n) = dim U(n) − dim O(n) = n2 − n(n − 1) 2 = n(n + 1) 2 . The graphs {Γ(Rn,φ) | φ∗ = φ ∈ Mn(R)} ⊂ Λ(n) define a chart at Rn ∈ Λ(n).
◮ Example (Arnold and Givental, 1985)
Λ(2)3 = {[x, y, z, u, v] ∈ R P4 | x2 + y2 + z2 = u2 + v2} = S2 × S1/{(x, y) ∼ (−x, −y)} .
SLIDE 16
16 The Lagrangian Grassmannian Λ(n) II.
◮ In view of the fibration
Λ(n) = U(n)/O(n) → BO(n) → BU(n) Λ(n) classifies real n-plane bundles β with a trivialisation δβ : C ⊗ β ∼ = ǫn of the complex n-plane bundle C ⊗ β.
◮ The canonical real n-plane bundle η over Λ(n) is
E(η) = {(L, ℓ) | L ∈ Λ(n), ℓ ∈ L} . The complex n-plane bundle C ⊗ η E(C ⊗ η) = {(L, ℓC) | L ∈ Λ(n), ℓC ∈ C ⊗R L} is equipped with the canonical trivialisation δη : C ⊗ η ∼ = ǫn defined by δη : E(C ⊗ η) ∼ = E(ǫn) = Λ(n) × Cn ; (L, ℓC) → (L, (p, q)) if ℓC = (p, q) ∈ C ⊗R L = L ⊕ iL = Cn .
SLIDE 17 17 The fundamental group π1(Λ(n)) I.
◮ Theorem (Arnold, 1967) The square of the determinant function
det2 : Λ(n) → S1 ; L = A(ıRn) → det(A)2 induces an isomorphism det2 : π1(Λ(n)) ∼ = π1(S1) = Z .
◮ Proof By the homotopy exact sequence of the commutative diagram of
fibre bundles SO(n)
det
- O(1) = S0
- SU(n)
- U(n)
- det U(1) = S1
z→z2
Λ(n)
det2 Λ(1) = S1
SLIDE 18
18 The fundamental group π1(Λ(n)) = Z II.
◮ Corollary 1 The function
π1(Λ(n)) → Z ; (ω : S1 → Λ(n)) → degree( S1
ω
Λ(n)
det2 S1 )
is an isomorphism.
◮ Corollary 2 The cohomology class
α ∈ H1(Λ(n)) = Hom(π1(Λ(n)), Z) characterized by α(ω) = degree( S1
ω
Λ(n)
det2 S1 ) ∈ Z
is a generator α ∈ H1(Λ(n)) = Z.
SLIDE 19 19 π1(Λ(1)) = Z in terms of line bundles
◮ Example The universal real line bundle η : S1 → BO(1) = R P∞ is the
infinite M¨
E(η) = R × [0, π]/{(x, 0) ∼ (−x, π)} → S1 = [0, π]/(0 ∼ π) . The complexification C ⊗ η : S1 → BU(1) = C P∞ is the trivial complex line bundle over S1 E(C ⊗ η) = C × [0, π]/{(z, 0) ∼ (−z, π)} → S1 = [0, π]/(0 ∼ π) . The canonical trivialisation δη : C ⊗ η ∼ = ǫ is defined by δη : E(C ⊗ η) ∼ = E(ǫ) = C × [0, π]/{(z, 0) ∼ (z, π)} = C × S1 ; [z, θ] → [zeiθ, θ] .
◮ The canonical pair (δη, η) represents
(δη, η) = 1 ∈ π1(Λ(1)) = π1(U(1)/O(1)) = Z , corresponding to the loop ω : S1 → Λ(1); e2iθ → L(θ).
SLIDE 20 20 The Maslov index for n-dimensional Lagrangians
◮ Unitary matrices can be diagonalized. For every A ∈ U(n) there exists
B ∈ U(n) such that BAB−1 = D(eiθ1, eiθ2, . . . , eiθn) is the diagonal matrix, with eiθj ∈ S1 the eigenvalues, i.e. the roots of the characteristic polynomial chz(A) = det(zIn − A) =
n
(z − eiθj) ∈ C[z] .
◮ Definition The Maslov index of L ∈ Λ(n) is
τ(L) =
n
(1 − 2θj/π) ∈ R with θ1, θ2, . . . , θn ∈ [0, π) such that ±eiθ1, ±eiθ2, . . . , ±eiθn are the eigenvalues of any A ∈ U(n) such that A(ıRn) = L.
◮ There are similar definitions of τ(L, L′) ∈ R and τ(L, L′, L′′) ∈ Z.
SLIDE 21
21 The Maslov cycle
◮ Corollary 2 above shows that the Maslov index α ∈ H1(Λ(n)) is the
pullback along det2 : Λ(n) → S1 of the standard form dθ on S1.
◮ Definition The Poincar´
e dual of α is the Maslov cycle m = {L ∈ Λ(n) | L ∩ ıRn = {0}} .
◮ In his 1967 paper Arnold constructs this cycle following ideas that
generalise the standard constructions on Schubert varieties in the Grassmannians of linear subspaces of Euclidean spaces, and proves that [m] ∈ H(n+2)(n−1)/2(Λ(n)) is the Poincar´ e dual of α ∈ H1(Λ(n)).
SLIDE 22
22 Lagrangian submanifolds of R2n I.
◮ An n-dimensional submanifold Mn ⊂ R2n is Lagrangian if for each
x ∈ M the tangent n-plane Tx(M) ⊂ Tx(R2n) = R2n is a Lagrangian subspace of (R2n, [ , ]).
◮ The tangent bundle TM : M → BO(n) is the pullback of the canonical
real n-plane bundle over Λ(n) E(η) = {(λ, ℓ) | λ ∈ Λ(n), ℓ ∈ λ} along the classifying map ζ : M → Λ(n); x → Tx(M), with TM : M ζ
Λ(n)
η
BO(n) . The complex n-plane bundle
C ⊗ TM = TCn|M has a canonical trivialisation.
◮ Definition The Maslov index of a Lagrangian submanifold Mn ⊂ R2n
is the pullback of the generator α ∈ H1(Λ(n)) τ(M) = ζ∗(α) ∈ H1(M) .
SLIDE 23
23 Lagrangian submanifolds of R2n II.
◮ Given a Lagrangian submanifold Mn ⊂ R2n let
π = proj| : Mn → Rn ; (p, q) → p be the restriction of the projection R2n = Rn ⊕ ıRn → Rn. The differentials of π are the differentiable maps dπx : Tx(M) → Tπ(x)(Rn) = Rn ; ℓ → p if ℓ = (p, q) ∈ Tx(M) ⊂ R2n with ker(dπx) = Tx(M) ∩ ıRn ⊆ ıRn.
◮ If π is a local diffeomorphism then each dπx is an isomorphism of real
vector spaces, with kernel Tx(M) ∩ ıRn = {0}.
◮ Definition The Maslov cycle of a Lagrangian submanifold M ⊂ R2n is
m = {x ∈ M | Tx(M) ∩ ıRn = {0}} .
◮ Theorem (Arnold, 1967) Generically, m ⊂ M is a union of open
submanifolds of codimension 1. The homology class [m] ∈ Hn−1(M) is the Poincar´ e dual of the Maslov index class τ(M) ∈ H1(M), measuring the failure of π : M → Rn to be a local diffeomorphism.
SLIDE 24
24 Symplectic manifolds
◮ Definition A manifold N is symplectic if it admits a 2-form ω which is
closed and non-degenerate.
◮ Examples S2, R2n = Cn = T∗Rn; the cotangent bundle T∗B of any
manifold B with a canonical symplectic form.
◮ Definition A submanifold M ⊂ N of a symplectic manifold is
Lagrangian if ω|M = 0 and each TxM ⊂ TxN (x ∈ M) is a Lagrangian subspace.
◮ Remark A symplectic manifold N is necessarily even dimensional
(because of the non-degeneracy of ω) and the dimension of a Lagrangian submanifold M is necessarily half the dimension of the ambient manifold.
◮ Examples If N is two dimensional, any one dimensional submanifold is
Lagrangian (why?). If N = T∗B for some manifold B with the canonical symplectic form, then each cotangent space T∗
bB is a
Lagrangian submanifold.
SLIDE 25 25 The Λ(n)-affine structure of Λ(V )
◮ Let V be a 2n-dimensional real vector space with nonsingular
symplectic form [ , ] : V × V → R and a complex structure ı : V → V such that V × V → R; (v, w) → [ıv, w] is an inner product. Let U(V ) be the group of automorphisms A : (V , [ , ], ı) → (V , [ , ], ı).
◮ For L ∈ Λ(V ) let O(L) = {A ∈ U(V ) | A(L) = L}. ◮ Proposition (Guillemin and Sternberg) (i) The function
U(V )/O(L) → Λ(V ) ; A → A(L) is a diffeomorphism.
◮ (ii) There exists an isomorphism fL : (R2n, [ , ], ı) ∼
= (V , [ , ], ı) such that fL(ıRn) = L.
◮ (iii) For any fL as in (ii) the diffeomorphism
fL : Λ(n) = U(n)/O(n) → Λ(V ) = U(V )/O(L) ; λ → fL(λ)
- nly depends on L, and not on the choice of fL.
◮ Thus Λ(V ) has a Λ(n)-affine structure: for any L1, L2 ∈ Λ(V ) there is
defined a difference element (L1, L2) ∈ Λ(n), with (L1, L1) = ıRn ⊂ R2n the imaginary lagrangian.
SLIDE 26
26 An application of the Maslov index I.
◮ For any n-dimensional manifold B the tangent bundle of the cotangent
bundle E = T(T∗B) → T∗B has each fibre a 2n-dimensional symplectic vector space. The symplectic form is given in canonical (or Darboux) coordinates as ω = dp ∧ dq (q ∈ B, p ∈ T∗
qB) . ◮ The bundle of Lagrangian planes Π : Λ(E) → T∗B is the fibre bundle
with fibres the affine Λ(n)-sets Π−1(x) = Λ(TxT∗B) of Lagrangians in the 2n-dimensional symplectic vector space TxT∗B.
SLIDE 27
27 An application of the Maslov index II.
◮ The projection π : T∗B → B induces the
vertical subbundle V = ker dπ ⊂ T(T∗B) , whose fibres are Lagrangian subspaces.
◮ The map
s : T∗B → Λ(E) ; x = (p, q) → Vx = ker(dπx : Tx(T∗B) → Tπ(x)(B)) is a section of the bundle of Lagrangian planes.
◮ For any other section r : T∗B → Λ(E) use the Λ(n)-affine structures of
the fibres Π−1(x) to define a difference map ζ = (r, s) : T∗B → Λ(n) ; x → (r(x), s(x))
SLIDE 28
28 An application of the Maslov index III.
◮ Let M be a Lagrangian submanifold of T∗B, dim M = dim B = n. ◮ If π|M : M → B is a local diffeomorphism then each
d(π|M)x : Tx(M) → Tπ(x)(B) is an isomorphism of vector bundles, with kernel Tx(M) ∩ Vx = {0}.
◮ The Maslov index measures the failure of π|M : M → B to be a local
diffeomorphism, in general.
◮ We let
EM = TM ⊂ T(T∗B) , Λ(EM) = Π−1(M) ⊂ Λ(E) so that Π| : Λ(EM) → M is a fibre bundle with the fibre over x ∈ M the Λ(n)-affine set (Π|)−1(x) = Λ(TxM).
SLIDE 29
29 An application of the Maslov index IV.
◮ The maps r, s|M : M → Λ(EM) defined by
r(x) = TxM , s|M(x) = Vx ⊂ Tx(T∗B) are sections of Π|. The difference map ζ = (r, s|M) : M → Λ(n) classifies the real n-plane bundle over M with complex trivialisation in which the fibre over x ∈ M is the Lagrangian (TxM, Vx) ∈ Λ(n).
◮ Definition The Maslov index class is the pullback along ζ of the
generator α = 1 ∈ H1(Λ(n)) = Z αM = ζ∗α ∈ H1(M) .
◮ Proposition The homology class [m] ∈ Hn−1(M) of the Maslov cycle
m = {x ∈ M | TxM ∩ Vx = {0}} , is the Poincar´ e dual of the Maslov index class αM ∈ H1(M).
◮ If the class [m] = αM ∈ Hn−1(M) = H1(M) is non-zero then the
projection π|M : M → B fails to be a local diffeomorphism.
SLIDE 30
30 An application of the Maslov index V.
◮ (Butler, 2007) This formulation of the Maslov index can be used to
prove that if Σ is an n-dimensional manifold which is topologically T n, and which admits a geometrically semi-simple convex Hamiltonian, then Σ has the standard differentiable structure.
SLIDE 31 31 References
◮ V.I.Arnold, On a characteristic class entering into conditions of
quantisation, Functional Analysis and Its Applications 1, 1-14 (1967)
◮ V.I.Arnold and A.B.Givental, Symplectic geometry, in Dynamical
Systems IV, Encycl. of Math. Sciences 4, Springer, 1-136 (1990)
◮ L.Butler, The Maslov cocycle, smooth structures and real-analytic
complete integrability, http://uk.arxiv.org/abs/0708.3157, to appear in
◮ V.Guillemin and S.Sternberg, Symplectic geometry, Chapter IV of
Geometric Optics, Math. Surv. and Mon. 14, AMS, 109-202 (1990)
